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astraxan [27]
3 years ago
6

How do you find the x intercepts of 3x^(5/3) - 4x^(7/3)

Mathematics
2 answers:
kaheart [24]3 years ago
7 0
Set the whole expression = to 0 and solve for x.

3x^(5/3) - 4x^(7/3) = 0.  Factor out x^(5/3):     x^(5/3) [3 - 4x^(2/3)] = 0

Then either x^(5/3) = 0, or 3 - 4x^(2/3) = 0.

In the latter case, 4x^(2/3) = 3. 

To solve this:  mult. both sides by x^(-2/3).  Then we have

4x^(2/3)x^(-2/3) = 3x^(-2/3),            or 4 = 3x^(-2/3).  It'd be easier to work with this if we rewrote it as

4           3
---  = --------------------
1            x^(+2/3)

Then 

4
---  = x^(-2/3).  Then, x^(2/3) = (3/4), and x = (3/4)^(3/2).  According to my     3                      calculator, that comes out to x = 0.65 (approx.)

Check this result!  subst. 0.65 for x in the given equation.  Is the equation then true?

My method here was a bit roundabout, and longer than it should have been.  Can you think of a more elegant (and shorter) solution?
Varvara68 [4.7K]3 years ago
7 0
To the risk of sounding redundant, as Altavistard already pointed out above

\bf 3x^{\frac{5}{3}}-4x^{\frac{7}{3}}=0\implies \stackrel{common~factor}{x^{\frac{5}{3}}}(3-4x^{\frac{2}{3}})=0\\\\
-------------------------------\\\\
x^{\frac{5}{3}}=0\implies \boxed{x=0}\\\\
-------------------------------\\\\
3-4x^{\frac{2}{3}}=0\implies 3=4x^{\frac{2}{3}}\implies (3)^3=\left(4x^{\frac{2}{3}}\right)^3
\\\\\\
27=64x^2\implies \cfrac{27}{64}=x^2\implies \sqrt{\cfrac{27}{64}}=x\implies \boxed{\cfrac{3\sqrt{3}}{4\sqrt{4}}=x}
\\\\\\
0.64951905283832898507\approx x
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Please help me i need to find the area of shaded region please help me​
Montano1993 [528]

Answer:

Part 1) A=60\ ft^2

Part 2) A=80\ cm^2

Part 3) A=96\ m^2

Part 4) A=144\ cm^2

Part 5) A=13\ m^2

Part 6) A=(49\pi -33)\ in^2

Step-by-step explanation:

Part 1) we know that

The shaded region is equal to the area of the complete rectangle minus the area of the interior rectangle

The area of rectangle is equal to

A=bh

where

b is the base of rectangle

h is the height of rectangle

so

A=(12)(7)-(8)(3)\\A=84-24\\A=60\ ft^2

Part 2) we know that

The shaded region is equal to the area of the complete rectangle minus the area of the interior square

The area of square is equal to

A=b^2

where

b is the length side of the square

so

A=(12)(8)-(4^2)\\A=96-16\\A=80\ cm^2

Part 3) we know that

The area of the shaded region is equal to the area of four rectangles plus the area of one square

so

A=4(4)(5)+(4^2)\\A=80+16\\A=96\ m^2

Part 4) we know that

The shaded region is equal to the area of the complete square minus the area of the interior square

so

A=(15^2)-(9^2)\\A=225-81\\A=144\ cm^2

Part 5) we know that

The area of the shaded region is equal to the area of triangle minus the area of rectangle

The area of triangle is equal to

A=\frac{1}{2}(b)(h)

where

b is the base of triangle

h is the height of triangle

so

A=\frac{1}{2}(6)(7)-(4)(2)\\A=21-8\\A=13\ m^2

Part 6) we know that

The area of the shaded region is equal to the area of the circle minus the area of rectangle

The area of the circle is equal to

A=\pi r^{2}

where

r is the radius of the circle

so

A=\pi (7^2)-(3)(11)\\A=(49\pi -33)\ in^2

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